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At least half of all spiral galaxies are warped: in the outer parts, one side of the disk curves up above the mean plane of the inner regions, while on the opposite side the disk bends downwards. The warp is most easily observed in neutral hydrogen, since the gas layer is the most extended component of the galaxy, but is sometimes seen also in the stellar disk; the warp of our own Milky Way is observed in atomic and molecular gas, and in both young and old stars. The near-ubiquity of warps presents a dynamical puzzle, since differential precession in the galactic potential should cause them to wind rapidly into a tight spiral form.
Hunter \& Toomre (1969) investigated an early suggestion (Lynden-Bell 1965) that the warp represented a discrete mode of vertical oscillation of the disk under its own gravity; they showed that an isolated galactic disk was unlikely to have such a mode, and could not retain a long-lived warp. Later, it became clear that much of the mass of a galaxy resides in a `dark halo'. The idea that warps might result when the galactic disk was tipped away from the equator of a flattened dark halo was put forward by Dekel \& Shlosman (1983), and by Toomre (1983), who pointed out that the misalignment could only persist if the disk had a discrete mode of bending within the halo potential. Sparke \& Casertano (1988) showed that discrete modes can indeed exist for realistic disk and halo parameters, with shapes resembling those of the observed warps. The disk is tilted at an angle to the symmetry plane of the halo, and differential precession is eliminated by the cohesive effect of self-gravity.
Recent time-dependent calculations (Hofner \& Sparke 1994) show that an initially perturbed disk settles towards the warped mode; the inner parts approach the mode shape first, while the outer disk is twisted into a leading spiral form, as found in observed warps. Settling in the outer parts of observed disks may require longer than a Hubble time. When a galactic bulge is included in the dynamical model (Pitesky 1993), new discrete modes can appear, in which the torque of the warped disk causes the bulge to become tipped relative to the plane of the inner galaxy.
